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  1. DELEUZE’S THE FOLD By Catherine Juyu Cheng 鄭如玉

  2. Two Floors • “the Baroque trait twists and turns its folds, pushing them to infinity, fold over fold, one upon the other. The Baroque fold unfurls all the way to infinity. First, the Baroque differentiates its folds in two ways, by moving along two infinities, as if infinity were composed of two stages or floors: the pleats of matter, and the folds in the soul” (The Fold 3).

  3. Cryptographer • If Descartes did not know how to get through the • labyrinth, it was because he sought its secret of continuity in rectilinear tracks, and the secret of liberty in a rectitude of the soul. He knew the inclension of the soul as little as he did the curvature of matter. • A 'cryptographer' is needed, someone who can at once account for nature and decipher the soul, who can peer into the crannies of matter and read into the folds of the soul.‘ (The Fold 3)

  4. Baroque House • Clearly the two levels are connected (this being why continuity rises • up into the soul). There are souls down below, sensitive, animal; and • there even exists a lower level in the souls. The pleats of matter surround • and envelop them. When we learn that souls cannot be furnished with • windows opening onto the outside, we must first, at the very least, • include souls upstairs, reasonable ones, who have ascended to the other • level ('elevation'). It is the upper floor that has no windows. It is a dark • room or chamber decorated only with a stretched canvas 'diversified by • folds,' as if it were a living dermis. Placed on the opaque canvas, these • folds, cords, or springs represent an innate form of knowledge, but when • solicited by matter they move into action. Matter triggers 'vibrations or • oscillations' at the lower extremity of the cords, through the intermediary • of 'some little openings' that exist on the lower level. Leibniz constructs a • great Baroque montage that moves between the lower floor, pierced with • windows, and the upper floor, blind and closed, but on the other hand • resonating as if it were a musical salon translating the visible movements below into sounds above. (The Fold 4)

  5. Baroque mathematical physics • Huygens develops a Baroque mathematical physics whose goal is curvilinearity. With Leibniz the curvature of the universe is prolonged according to three other fundamental notions: • 1. the fluidity of matter, • 2. the elasticity of bodies, • 3. motivating spirit as a mechanism. (The Fold 4)

  6. Curvilinear • First, matter would clearly not be extended following a twisting line. Rather, it would follow a tangent.' But the universe appears compressed by an active force that endows matter with a curvilinear or spinning movement, following an arc that ultimately has no tangent. And the infinite division of matter causes compressive force to return all portions of matter to the surrounding areas, to the neighboring parts that bathe and penetrate the given body, and that determine its curvature. Dividing endlessly, the parts of matter form little vortices in a maelstrom, and in these are found even more vortices, even smaller, and even more are spinning in the concave intervals of the whirls that touch one another. (The Fold 4)

  7. No Void • Matter thus offers an infinitely porous, spongy, or cavernous texture without emptiness, caverns endlessly contained in other caverns: no matter how small, each body contains a world pierced with irregular passages, surrounded and penetrated by an increasingly vaporous fluid, the totality of the universe resembling a 'pond of matter in which there exist different flows and waves.' (The Fold 5)

  8. 1. The Fluidity of Matter • 1. Descartes's error probably concerns what is to be • found in different areas. He believed that the real distinction between • parts entailed separability. What specifically defines an absolute fluid is the absence of coherence or cohesion; that is, the separability of parts, which in fact applies only to a passive and abstract matter. • 2. According to Leibniz, two parts of really distinct matter can be inseparable, as shown not only by the action of surrounding forces that determine the curvilinear movement of a body but also by the pressure of surrounding forces that determine its hardness (coherence, cohesion) or the inseparability of its parts. Thus it must be stated that a body has a degree of hardness as well as a degree of fluidity, or that it is essentially elastic, the elastic force of bodies being the expression of the active compressive force exerted on matter • (The Fold 5-6) • (99)trompe l’oeil

  9. The Elasticity of Bodies vs. A Spirit in Matter • 2.The Elasticity of Bodies: a flexible or an elastic body still has cohering parts that form a fold, such that they are not separated into parts of parts but are rather divided to infinity in smaller and smaller folds that always retain a certain cohesion. (The Fold 6) • 3. A Spirit in Matter: If the world is infinitely cavernous, if worlds exist in the tiniest bodies, it is because everywhere there can be found 'a spirit in matter,‘ (The Fold 7).

  10. endogenous vs. exogenous • The lower level or floor is thus also composed of organic matter. An organism is defined by endogenous folds, while inorganic matter has exogenous folds that are always determined from without or by the surrounding environment. Thus, in the case of living beings, an inner • formative fold is transformed through evolution, with the organism's development. Whence the necessity of a preformation. (The Fold 7)

  11. Pond Elastic forces (Waves) 1. Pond Plastic Forces (Fish Swarm) 2. Elastic Forces mechanic (water, wind, ore) 3. Plastic forces machinelike 4. Matter is folded twice, once under elastic forces, a second time under plastic forces, but one is not able to move from the first to the second. (9) (The Fold 9)

  12. Einfalt vs. Zweifalt • With preformism, an organic fold always ensues from another fold, at least on the inside from a same type of organization: every fold originates from a fold, plica ex plica. if Heideggerian terms can be used, we can say that the fold of epigenesis is an Einfalt, or that it is the differentiation of an undifferentiated, but that the fold from prefomation is a Zweifalt, not a fold in two--since every fold can only be thus but a fold-of-two, and entre-deux, something "between" in the sense that a difference is being differentiated (10).

  13. 1. Plastic forces of matter act on masses, but they submit them to real unities that they take for granted. They make an organic synthesis, but assume the soul as the unity of synthesis, or as the 'immaterial principle of life.' (11) • 2. The Soul: In the Baroque the soul entertains a complex relation with the body. Forever indissociable from the body, it discovers a vertiginous animality that gets it tangled in the pleats of matter, but also an organic or cerebral humanity (the degree of development) that allows it to rise up, and that will make it ascend over all other folds. (11)

  14. The Reasonable Soul • 1. Bare entelechies • 2. animal souls • 3. reasonable minds • The reasonable soul is free, like a Cartesian diver, to fall back down at death and to climb up again at the last judgment. As Leibniz notes, the tension is between the collapse and the elevation or ascension that in different spots is breaching the organized masses. We move from funerary figures of the Basilica of Saint Laurence to the figures on the ceiling of Saint Ignatius. (11)

  15. Funerary figures of the Basilica of Saint Laurence p.11

  16. The figures on the ceiling ofSaint Ignatius.

  17. No Windows • Hence the need for a second floor is everywhere affirmed to be strictly • metaphysical. The soul itself is what constitutes the other floor or the • inside up above, where there are no windows to allow entry of influence • from without. Even in a physical sense we are moving across outer • material pleats to inner animated, spontaneous folds. These are what we • must now examine, in their nature and in their development. Everything • moves as if the pleats of matter possessed no reason in themselves. It is • because the Fold is always between two folds, and because the between two-folds seems to move about everywhere: Is it between inorganic • bodies and organisms, between organisms and animal souls, between • animal souls and reasonable souls, between bodies and souls in general? • (The Fold 13)

  18. Chapter 2 • 1. Inflection • 2. Vectorial, projective, infinite variation • 3. Three kinds of points as three kinds of singularities: • (1) physical point (the point of inflection) • (2) mathematical point (the point of position)(3) metaphysical point (the point of inclusion)

  19. Paul Klee: inflection is the authentic atom, the elastic point. • The first draws the inflection. • The second shows that no exact and unmixed figure can exist. As Leibniz stated, there can never be 'a straight line without curves intermingled,' • 3. The third marks the convex side with • shadow, and thus disengages • concavity and the axis of its curve, that • now and gain changes sides from the • point of inflection. Bernard Cache • defines inflection — or the point of • inflection.

  20. Paul Klee (14)

  21. Kandinsky: A Cartesian, for whom angles are firm (4)

  22. Bernard Cache • Thus inflection is the pure Event of the line or of the point, the Virtual. (15) • Cache’s three transformations: • A. Vectorial with tangent plane of reflection, work according to optical laws, transforming inflection at a turning point (15)

  23. B. Projective such transformations convey the projection, on external space, of internal spaces defined by "hidden parameters" and variables or singularities of potential. Rene Thom transformations refer in this sense to a morphology of living matter, providing seven elementary events: the fold; the crease, the dovetail, the butterfly, the hyperbolic, elliptical and parabolic umbilicus (16)

  24. Rene Thom

  25. C. Infinite variation or infinitely variable curve • the inflection in itself cannot be separated from an infinite variation or an infinitely variable curve. Such is Koch’s curve, obtained by means of rounding angles, according to Baroque requirements, by making them proliferate according to a law of homothesis. The curve passes through an infinite number of angular points and never admits a tangent at any of these points. It envelops an infinitely cavernous or porous world, constituting more than a line and less than a surface (Mandelbrot’s fractal dimension as a fractional or irrational number, a nondimension, an interdimension). It is no longer possible to determine an angular point between two others, no matter how close one is to the other, but there remains the latitude to always add a detour by making each interval the site of a new folding. That is how we go from fold to fold and not from point to point, and how every contour is blurred to give definition to the formal powers of the raw material, which rise to the surface and are put forward as so many detours and supplementary folds. Transformation of inflection can no longer allow for either symmetry or the favored plane of projection. It becomes vortical and is produced later; deferred, rather than prolonged or proliferating (16-17).

  26. Mandelbrot’s fractal dimension Koch’s Curve

  27. The Irrational Number • 1. The definition of Baroque mathematics is born with Leibniz—the irrational number is the common limit of two convergent series, of which one has no maximum and the other no minimum…Theirrational number implies the descent of a circular arc on the straight line of rational points, and exposes the latter as a false infinity, a simple undefinite that includes an infinity of lacunae; that is why the continuous is a labyrinth that cannot be represented by a straight line. The straight line always has to be intermingled with curved lines. (17)

  28. Point-Fold • 2. Between the two points A and B — no matter in what proximity they may be — there always remains the possibility for carrying out the right isosceles triangle, whose hypotenuse goes from A to B, and whose summit, C, determines a circle that crosses the straight line between A and B. The arc of the circle resembles a branch of inflection, an element of the labyrinth, that from an irrational number, at the meeting of the curved and straight lines, produces a point-fold. (18)

  29. Objectile • The new object is an objectile. the new status of the object no longer refers its condition to a spatial mold-in other words, to a relation of form-matter-but to a temporal modulation that implies as much the beginnings of a continuous variation of matter as a continuous development of form. In modulation "a pause never intervenes for withdrawal; a modulator is a continuous temporal mold...Molding amounts to modulating in a definitive way; modulating is molding in a continuous and perpetually variable fashion." the object here ismanneristic, not essentializing: it becomes an event (19).

  30. Superject • The transformation of the object refers to a correlative transformation of the subject (19-20)Such is the basis of perspectivism, which does not mean a dependence in respect to a pregiven or defined subject; to the contrary, a subject will be what comes to the point of view, or rather what remains in the point of view. That is why the transformation of the object refers to a correlative transformation of the subject: the subject is not a subject but, as Whitehead says, a 'superject.' Just as the object becomes objectile, the subject becomes a superject (19-20).

  31. Point of view on a variation now replaces the center of a figure or a configuration. The most famous example is that of conic sections, where the point of the cone is the point of view to which the circle, the ellipse, the parabola, and the hyperbola are related as so many variants that follow the incline of the section that is planned ('scenographies') (20-21). Ambiguous sign • This objectile or projection resembles an unfolding. But unfolding is no more the contrary of foldings than an invariant would be the contrary of variation. It is an invariant of transformation. Leibniz will designate it by an 'ambiguous sign." (21). (not coordinates)

  32. Desargues called the relation or the law enveloped by a variation 'involution' (for example, a triangle that is supposed to turn around an axis, the dispositions of the points defined on the axis by the projection of three summits and by the prolongation of the three sides)." (21)

  33. The Soul and Inflection • A soul always includes what it apprehends from its point of view, in other words, inflection. Inflection is an ideal condition or a virtuality that currently exists only in the soul that envelops it. Thus, the soul is what has folds and is full of folds (22)

  34. Folds are in the Soul • Folds are in the soul and authentically exist only in the soul. That is already true for innate ideas: they are pure virtualities, pure powers whose act consists in habitus or arrangements (folds) in the soul, and whose completed act consists of an inner action of the soul (an internal deployment). But this is no less true for the world: the whole world is only a virtuality that currently exists only in the folds of the soul which convey it, the soul implementing inner pleats through which it endows itself with a representation of the enclosed world. We are moving from inflection to inclusion in a subject, as if from the virtual to the real, inflection defining the fold, but inclusion defining the soul or the subject, that is, what envelops the fold, its final cause and it completed act (23).

  35. Three kinds of points as three kinds of singularities: • (1) physical point (the point of inflection) : what runs along inflection or is the point of inflection itself: it is neither an atom nor a Cartesian point, but an elastic or plastic point-fold. • (2) mathematical point (the point of position): loses exactitude in order to become a position, a site, a focus, a place, a point of conjunction of vectors of curvature or, in short , point of view...pure extension will be the continuation or diffusion of the point • (3) metaphysical point (the point of inclusion)—then soul or the subject. It is what occupies the point of view, It is what is projected in point of view. Thus the soul is not in a body in a point, but is itself a higher point and of another nature, which corresponds with the point of view.(projection) (23)

  36. The Monad • Everyone knows the name that Leibniz ascribes to the soul or to the subject as a metaphysical point: the monad. • The world is the infinite curve that touches at an infinity of points an infinity of curves, the curve with a unique variable, the convergent series of all series (24). • As an individual unit each monad includes the whole series; hence it conveys the entire world, but does not express it without expressing more clearly a small region of the world, a “subdivision,” a borough of the city, a finite sequence (25).

  37. Adam and the World • We move from inflections of the world to inclusion in its subjects: how • can this be possible since the world only exists in subjects that include it? • In this respect the first letters to Arnauld specify the conciliation of the • two essential propositions. On the one hand, the world in which Adam • committed sin exists only in Adam the sinner (and in all other subjects • who make up this world). On the other hand, God creates not only Adam the sinner but also the world in which Adam has committed sin. In other words, if the world is in the subject, the subject is no less for the world. God produces the world 'before' creating souls since he creates them for this world that he invests in them. In this very way the law of infinite seriality, the 'law of curvatures,' no longer resides in the soul, although seriality may be the soul, and although curvatures may be in it. (The Fold 25)

  38. The Monad and the World • It is in this sense too that the soul is a 'production,' a 'result.' The soul results from the world that God has chosen. Because the world is in the monad, each monad includes every series of the states of the world; but, because the monad is for the world, no one clearly contains the 'reason‘ of the series of which they are all a result, and which remains outside of them, just like the principle of their accord. We thus go from the world to the subject, at the cost of a torsion that causes the monad to exist currently only in subjects, but that also makes subjects all relate to this world as if to the virtuality that they actualize. When Heidegger tries to surpass intentionality as an overly empirical determination of the subject's relation to the world, he envisions how Leibniz's formula of the monad without windows is a way to get past it, since the Dasein, he says, is already open at all times and does not need windows by which an opening would occur to it. (26)

  39. But in that way he mistakes the condition of closure or concealment enunciated by Leibniz; that is, the determination of a being-for the world instead of a being-in the world. Closure is the condition of being for the world. The condition of closure holds for the infinite opening of the finite: it 'finitely represents infinity.' It gives the world the possibility of beginning over and again in each monad. The world must be placed in the subject in order that the subject can be for the world. This is the torsion that constitutes the fold of the world and of the soul. And it is what gives to expression its fundamental character: the soul is the expression of the world (actuality), but because the world is • what the soul expresses (virtuality). Thus God creates expressive souls only because he creates the world that they express by including it: from inflection to inclusion (26).

  40. The Monad • Deleuze treats Leibniz as the philosopher of the Baroque and appropriates Leibniz’s concept of monad to delineate his own concept of subjectivity. The great Baroque montage drawn by Leibniz vividly reveals the characteristics of monad: Leibniz constructs a great Baroque montage with two floors (the upper belongs to the soul while the lower floor the matter). The important thing is that there is “a correspondence between … the pleats of matter and the folds in the soul” (The Fold 4). The following five points are the most important characteristics of Leibniz’s monad: • First, Leibniz “ascribes to the soul or to the subject as a metaphysical point: the monad” (The Fold 23). • Second, in The Fold, Deleuze tries to reconcile Leibniz and Locke by means of incorporating Baroque art, especially architecture, into the illustration of the monad. • Third, “each monad includes the whole series” (The Fold 25) and thus conveys the whole world. • Fourth, Leibniz further claims that “God determines the nature of each monad so that its state will be coordinated in a pre-established harmony without the need for interference” (Thomson 54). • Fifth, Leibniz establishes the monad as “absolute interiority and treats “the outside as an exact reversion, or ‘membrane,’ of the inside” (Badiou 61).

  41. Chapter 3 • The Traits of the Baroque • 1. The Fold • 2. The inside and the outside • 3. The high and low • 4. The unfold • 5. Textures • 6. The paradigm

  42. No windows, no models on its outside • Monads “have no windows, by which anything could come in or go out.” They have neither 'openings nor doorways." We run the risk of • understanding the problem vaguely if we fail to determine the situation. • A painting always has a model on its outside; it always is a window. If a • modern reader thinks of a film projected in darkness, the film has • nonetheless been projected. Then what about invoking numerical images • issuing from a calculus without a model? Or, more simply, the line with • infinite inflection that holds for a surface, like the lines of Pollock's or • Rauschenberg's painting? More exactly, in Rauschenberg's work we • could say that the surface stops being a window on the world and now • becomes an opaque grid of information on which the ciphered line is • written. The painting-window is replaced by tabulation, the grid on • which lines, numbers, and changing characters are inscribed (the • objectile)(27)

  43. Pollock’s Painting Rauschenberg’s Painting

  44. camera obscura • Leibniz is endlessly drawing up linear and numerical tables. With them he decorates the inner walls of the monad. Folds replace holes. The dyad of the city-information table is opposed to the system of the window-countryside. Leibniz's monad would be just such a grid — or better, a room or an apartment — completely covered with lines of variable inflection. This would be the camera obscura of the New Essays, furnished with a stretched canvas diversified by moving, living folds. Essential to the monad is its dark background: everything is drawn out of it, and nothing goes out or comes in from the outside. (27)

  45. camera obscura • First of all, the camera obscura has only one small aperture high up through which light passes, then through the relay of two mirrors it projects on a sheet the objects to be drawn that cannot be seen, the second mirror being tilted according to the position of the sheet (28).

  46. Camera Obscura

  47. Camera obscura